.TH "strsna.f" 3 "Wed Oct 15 2014" "Version 3.4.2" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME strsna.f \- .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBstrsna\fP (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO)" .br .RI "\fI\fBSTRSNA\fP \fP" .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine strsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, real, dimension( ldt, * )T, integerLDT, real, dimension( ldvl, * )VL, integerLDVL, real, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )S, real, dimension( * )SEP, integerMM, integerM, real, dimension( ldwork, * )WORK, integerLDWORK, integer, dimension( * )IWORK, integerINFO)" .PP \fBSTRSNA\fP .PP \fBPurpose: \fP .RS 4 .PP .nf STRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q orthogonal). T must be in Schur canonical form (as returned by SHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. .fi .PP .RE .PP \fBParameters:\fP .RS 4 \fIJOB\fP .PP .nf JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (SEP); = 'B': for both eigenvalues and eigenvectors (S and SEP). .fi .PP .br \fIHOWMNY\fP .PP .nf HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT. .fi .PP .br \fISELECT\fP .PP .nf SELECT is LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the eigenpair corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select condition numbers corresponding to a complex conjugate pair of eigenvalues w(j) and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix T. N >= 0. .fi .PP .br \fIT\fP .PP .nf T is REAL array, dimension (LDT,N) The upper quasi-triangular matrix T, in Schur canonical form. .fi .PP .br \fILDT\fP .PP .nf LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). .fi .PP .br \fIVL\fP .PP .nf VL is REAL array, dimension (LDVL,M) If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by SHSEIN or STREVC. If JOB = 'V', VL is not referenced. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. .fi .PP .br \fIVR\fP .PP .nf VR is REAL array, dimension (LDVR,M) If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by SHSEIN or STREVC. If JOB = 'V', VR is not referenced. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (MM) If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of S are set to the same value. Thus S(j), SEP(j), and the j-th columns of VL and VR all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected). If JOB = 'V', S is not referenced. .fi .PP .br \fISEP\fP .PP .nf SEP is REAL array, dimension (MM) If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of SEP are set to the same value. If the eigenvalues cannot be reordered to compute SEP(j), SEP(j) is set to 0; this can only occur when the true value would be very small anyway. If JOB = 'E', SEP is not referenced. .fi .PP .br \fIMM\fP .PP .nf MM is INTEGER The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M. .fi .PP .br \fIM\fP .PP .nf M is INTEGER The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension (LDWORK,N+6) If JOB = 'E', WORK is not referenced. .fi .PP .br \fILDWORK\fP .PP .nf LDWORK is INTEGER The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (2*(N-1)) If JOB = 'E', IWORK is not referenced. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor:\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBDate:\fP .RS 4 November 2011 .RE .PP \fBFurther Details: \fP .RS 4 .PP .nf The reciprocal of the condition number of an eigenvalue lambda is defined as S(lambda) = |v**T*u| / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of T corresponding to lambda; v**T denotes the transpose of v, and norm(u) denotes the Euclidean norm. These reciprocal condition numbers always lie between zero (very badly conditioned) and one (very well conditioned). If n = 1, S(lambda) is defined to be 1. An approximate error bound for a computed eigenvalue W(i) is given by EPS * norm(T) / S(i) where EPS is the machine precision. The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose T = ( lambda c ) ( 0 T22 ) Then the reciprocal condition number is SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) where sigma-min denotes the smallest singular value. We approximate the smallest singular value by the reciprocal of an estimate of the one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to be abs(T(1,1)). An approximate error bound for a computed right eigenvector VR(i) is given by EPS * norm(T) / SEP(i) .fi .PP .RE .PP .PP Definition at line 264 of file strsna\&.f\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.